cascaded systems analysisbml.pusan.ac.kr/lectureframe/lecture/graduates/image... · 2020-03-17 ·...
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Cascaded Systems Analysis
Ho Kyung [email protected]
Pusan National University
Medical Imaging Detectors
Noise Transfer
Image quality basically depends on # quanta interacting w/ an imaging system
Information from the quanta detected by a detector should be expressed as a final image without any loss; but
• Complex transfer mechanism: the imaging system consists of multiple processes from detecting x‐ray quanta to displaying the final image
• Inefficient transfer: un‐optimally designed system can include additional factors degrading image quality
Cascaded‐systems analysis
• Represent the complex system as a cascade of elementary stages
‒ Quantum amplification
‒ Deterministic blurring
‒ Quantum scattering
• Use transfer theory to describe the transfer of signal/noise thru the system
• Predict system performance based on design parameters
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Quantum amplification
Quantum gain or selection
Conversion of quanta from one form to another
• e.g. X‐ray quanta to optical quanta in a scintillator
• 𝑞 𝐫 𝑔𝑞 𝐫
‒ 𝑞 𝐫 = quantum image
‒ 𝑔 = gain; RV characterized by a mean 𝑔 & variance 𝜎
Mean number of quanta
• 𝑞 �̅�𝑞
NPS
• NPS 𝐤 �̅� NPS 𝐤 𝑞 𝜎
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Binomial selection
Special case of the quantum amplification process (𝑔 1) Describe the quantum efficiency of a detector
• Each quantum is either detected (or transferred thru this selection stage w/ prob. �̅�) or not (w/ prob. 1 �̅�)
• 𝑔 = RV having a value of 1 or 0 only
• Variance 𝜎 �̅� 1 �̅�
Noise transfer
• NPS 𝐤 �̅� NPS 𝐤 𝑞 �̅� 1 �̅� �̅� NPS 𝐤 𝑞 𝑞 �̅�
‒ If NPS 𝐤 ≪ 𝑞 (significant correlated noise) ⇒ NPS 𝐤 𝑞 𝑔 1 𝑔‒ If NPS 𝐤 𝑞 (uncorrelated input quanta), ⇒ NPS 𝐤 𝑞 𝑔
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Uncorrelated noiseCorrelated noise
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Deterministic blur
Blur resulting from a convolution (linear filtering) of the input w/ a PSF
Input signal redistribution w/ a weighting given by the PSF
• 𝑑 𝐫 𝑞 𝐫 ∗ psf 𝐫
Signal
• �̅� 𝑞‒ Analog output
Noise transfer
• NPS 𝐤 NPS 𝐤 MTF 𝐤
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Quantum scatter
Random relocation of an input quantum to a new location w/ a prob. of PSF
Redistribution w/ probabilities
• Note that the deterministic blur is the redistribution of signal by weights
• 𝑞 𝐫 𝑞 𝐫 ∗ psf 𝐫
Signal
• 𝑞 𝑞
NPS
• NPS 𝐤 NPS 𝐤 𝑞 MTF 𝐤 𝑞
‒ NPS 𝐤 NPS 𝐤 MTF 𝐤 𝑞 1 MTF 𝐤
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Uncorrelated noiseCorrelated noise
Deterministic blur Additional noise
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7Cunningham et al. IEEE EMBC & CMBEC (1995)
Cascade of elementary stages
Amplification + scattering
• e.g. Conversion of x‐ray photons to light quanta & their scatter w/i a scintillator
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Hypothetical detector
Incident x‐ray quanta
• 𝑞• NPS 𝑢 𝑞
1) Selection of interacting x rays
• 𝑞 𝑞 𝛼• NPS 𝑢 𝑞 𝛼
2) Conversion from x rays to light
• 𝑞 𝑞 𝛼𝑚
• NPS 𝑢 𝑞 𝛼𝑚 1
3) Spatial spreading of light
• 𝑞 𝑞 𝛼𝑚
• NPS 𝑢 𝑞 𝛼𝑚 1 MTF 𝑢 𝑞 𝛼𝑚
4) Selection of light quanta
• 𝑞 𝑞 𝛼𝑚𝛽
• NPS 𝑢 𝑞 𝛼𝑚 𝛽 1 MTF 𝑢 𝑞 𝛼𝑚𝛽
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Quantum‐Sink Analysis
DQE of a detector system w/ 𝑀 Poisson gain stages
• e.g. Photo‐multiplier tube
• DQE 𝑢⋯
⋯⋯
‒ P ∏ 𝑔
‒ P = # quanta at the 𝑗th stage normalized to # input quanta
‒ Called the "particle‐based analysis"
• DQE degrades if 1 or P 1
‒ System w/ a "quantum sink" at the 𝑗th stage‒ P = quantum efficiency of the detector
‒ P 1 always; called the "primary quantum sink"
• To avoid a secondary quantum sink, P 10‒ Not predict a secondary quantum sink at non‐zero spatial frequencies
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Quantum‐accounting diagram
Plot P as a function of the stage number 𝑗
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Fourier‐based analysis
• 2nd‐order statistics
• DQE 𝑢⋯
⋯ ⋯
• Poisson excess: 𝜀 1
‒ Poisson amplification (𝜎 𝑔 ): 𝜀 0
‒ Deterministic gain (a gain w/ no random variability, 𝜎 0): 𝜀 1
• Each stage either be an amplification or a scattering (not both)
‒ Amplification stage: MTF 𝑢 1 (gain w/ no blur)
‒ Scattering stage: 𝑔 1 & 𝜀 1 (blur w/ no gain)
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If 𝜀 ≪ 1;
• DQE 𝑢⋯
⋯ ⋯⋯
• P 𝑢 ∏ �̅� MTF 𝑢‒ P 𝑢 ≲ 1 degrades the DQE
‒ P 𝑢 1 has no effect on the DQE
To avoid a secondary quantum sink, P 𝑢 10 & P 0 100• Assumption: MTF 𝑢 0.33, where 𝑢 = max. freq. of interest
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𝛼 0.5
�̅� 10
𝛽 0.02
Minor 2ndary q sink
Dominant 2ndary q sink
1.0 1.1 1.25 1.4 1.7
2.0 2.5 3.3 5.0 10.0
Digital Metrics
Digital detector = an array of (discrete) detector elements
Detector element ('del' or 'pixel')
• Produce a signal proportional to # quanta interacting in the element
• Regarded as a spatial integrator of image quanta
‒ Integration of quanta in a detector element is represented as a convolution integral
• (1‐D) 𝑑 𝑘 𝑞 𝑥 d𝑥/
/
• 𝑑 𝑘 𝑞 𝑥 ∏ d𝑥
• 𝑑 𝑘𝑞 𝑥 ∗ ∏ 𝑑 𝑥 |
𝑑 𝑥 = presampling detector signal
• Analog (continuous) sample function
• Detector signal for all possible detector‐element positions
• Evaluation at 𝑥 𝑛𝑥 , or 𝑑 𝑥 | gives the detector signal
• 𝑑 𝑥 𝑘𝑞 𝑥 ∗ ∏ 𝑥/𝑎 ⇔ 𝐷 𝑢 𝑄 𝑥 T 𝑢
• Aperture MTF MTF 𝑢 sinc 𝜋𝑎 𝑢
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Digital MTF
Sampling
• The process of evaluating a function
‒ 𝑑 𝑘𝑞 𝑥 ∗ ∏ 𝑑 𝑥 |
Evaluating 𝑑 𝑥 at 𝑥 𝑛𝑥 for all 𝑛:• 𝑑 𝑥 𝑑 𝑥 ∑ 𝛿 𝑥 𝑛𝑥 ∑ 𝑑 𝛿 𝑥 𝑛𝑥
‒ Infinite train of 𝛿 functions scaled by the detector values 𝑑
• ℱ 𝑑 𝑥 𝐷 𝑢 ∗ ∑ 𝛿 𝑢
‒ Aliases of 𝐷 𝑢 at spacings of 𝑢 1/𝑥‒ Aliasing (overlap of aliases): distortion of the image signal at 𝑢 𝑢 1/ 2𝑥
Effect of the digital detector
• Attenuate spatial frequencies by the presampling MTF, MTF 𝑢‒ If 𝛾 1 (𝑎 𝑥 ), MTF 𝑢 sinc 𝜋𝑥 𝑢 ; the 1st zero at 𝑢 1/2𝑥
‒ If 𝛾 1 (𝑎 𝑥 ), MTF 𝑢 sinc 𝜋𝑎 𝑢 ; increasing bandwidth & more aliasing
• Introduce aliasing
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AliasesAliasing
Digital NPS
Discrete values 𝑑 : neither WSS nor WSCS random processes
Presampling detector signal 𝑑 𝑥 : WSS random process
Sampled signal 𝑑 𝑥 (an array of 𝛿‐functions scaled by 𝑑 ): WSCS random process
• NPS 𝑢 NPS 𝑢 ∗ ∑ 𝛿 𝑢 NPS 𝑢 ∑ NPS 𝑢
• Note again that the sampling theorem states that frequencies above the cut‐off frequency 𝑢1/ 2𝑥 cannot be represented w/ sample obtained w/ a uniform sampling frequency of 𝑢1/𝑥
Truncation to the cut‐off freq. range (= convolution w/ a sinc fnt = sinc interpolation)
• Estimate of 𝑑 𝑥 : 𝑑 𝑥 ∑ 𝑑 sinc 𝜋 𝑑 𝑥 ∗ sinc 𝜋𝑥 𝑢
• NPS 𝑢 NPS 𝑢 𝑥 ∏ 𝑥 𝑢
• 𝑑 𝑥 𝑑 𝑥 only if there is no aliasing of the presampling NPS
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Aliases centered at 𝑢 𝑛/𝑥
Fundamental presampling NPS
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Digital NPS
• NPS 𝑢/
E DFT ∆𝑑
‒ ∆𝑑 𝑑 E 𝑑‒ Numerical estimate of the NPS of 𝑑 𝑥
• NPS 𝑢 NPS 𝑢 𝑥 NPS 𝑢 NPS 𝑢 ∑ NPS 𝑢
‒ For 𝑢 𝑚/𝑁𝑥 & 𝑚 1
2‐D digital NPS
• NPS 𝑢, 𝑣 E DFT ∆𝑑 ,
1‐D NPS of 2‐D noise process represented by a digital image
• NPS 𝑢 E DFT ∑ ∆𝑑 ,
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Undesirable aliases
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Noise variance from NPS
• 𝜎 NPS 𝑢 d𝑢 𝑥 NPS 𝑢 d𝑢/
/
• 𝜎 NPS 𝑢 ∑ NPS 𝑢 d𝑢/
/
Pixel variance
• 𝜎 ∑ ∆𝑑
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Digital NEQ & DQE
Digital NEQ is defined only for 𝑢 𝑢 1/ 2𝑥
• NEQ 𝑞, 𝑢
‒ For 𝑢 𝑚/𝑁𝑥 & 𝑚 1
‒ Note that MTF 𝑢 includes the "aperture" MTF
For linear digital systems
• NEQ 𝑞, 𝑢/
• NEQ 𝑞, 𝑢∑
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Digital DQE
• DQE 𝑞, 𝑢,
∑
Signal aliasing
• Viewed as a form of image noise, hence resulting in additional artifacts & image degradation
• Not included in NEQ & DQE because being neither WSS nor WSCS
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Cascaded Model for A Hypothetical FPD
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E 𝑞 𝑞
𝑊 𝐤 𝑞
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Stage 1 [mm‐2]: selection of x‐ray quanta that interact in phosphor
• Quantum selection stage
‒ 𝛼 = RV having 0 & 1 only w/ an expected value 𝛼 (known as quantum efficiency)
• Quantum image: 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼• Expected value: 𝑞 𝛼𝑞• NPS: 𝑊 𝑢, 𝑣 𝛼𝑞
Stage 2 [mm‐2]: conversion to optical quanta in screen [mm‐2]
• 𝑚 optical quanta per interaction w/ a variance 𝜎‒ 𝜎 accounts for all variations in the conversion gain, incl. Swank noise & a polychromatic x‐ray beam
• 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼𝑚• 𝑞 𝛼𝑚𝑞
• 𝑊 𝑢, 𝑣 𝛼𝑚 𝑞 1 𝛼𝑚𝑞
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Stage 3 [mm‐2]: scattering of optical quanta in phosphor
• Optical quanta scatter w/ the same (normalized) psf 𝑥, 𝑦‒ Neglecting variable interacting depths
• 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦• 𝑞 𝛼𝑚𝑞
• 𝑊 𝑢, 𝑣 𝛼𝑚 𝑞 1 T 𝑢, 𝑣 𝛼𝑚𝑞
Stage 4 [mm‐2]: selection of optical quanta that interact
• A fraction 𝛽 of all optical quanta interacts somewhere in the photodiode array
‒ Including the coupling efficiency & quantum efficiency
• 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦 𝛽• 𝑞 𝛼𝑚𝛽𝑞
• 𝑊 𝑢, 𝑣 𝛼𝑚 𝛽 𝑞 1 T 𝑢, 𝑣 𝛼𝑚𝛽𝑞
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Stage 5 [mm2]: spatial integration of interacting optical quanta in elements
• Deterministic blur stage
‒ Integral of 𝑞 𝑥, 𝑦 over 𝑎 & 𝑎 ⇒ detector presampling signal
‒ 𝑘 = scaling factor relating 𝑞 & 𝑑
• 𝑑 𝑥, 𝑦 𝑘 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦 𝛽 ∗ ∏ ,
• �̅� 𝑘𝑎 𝑎 𝛼𝑚𝛽𝑞
• 𝑊 𝑢, 𝑣 𝑘 𝑎 𝑎 𝛼𝑚 𝛽 𝑞 1 T 𝑢, 𝑣 𝛼𝑚𝛽𝑞 sinc 𝜋𝑎 𝑢 sinc 𝜋𝑎 𝑣
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Stage 6 [mm2]: output from discrete detector elements
• Sampling stage w/ pitches of 𝑥 & 𝑦
• 𝑑 𝑥, 𝑦 𝑘 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦 𝛽 ∗ ∏ , ∑ ∑ 𝛿 𝑥 𝑛 𝑥 , 𝑦 𝑛 𝑦
• E 𝑑 𝑥, 𝑦 E 𝑑 , 𝑘𝑎 𝑎 𝛼𝑚𝛽𝑞
• 𝑊 𝑢, 𝑣 𝑊 𝑢, 𝑣 ∑ ∑ 𝑊 𝑢 , 𝑣
• 𝑊 𝑢, 𝑣 𝑊 𝑢, 𝑣 ∑ ∑ 𝑊 𝑢 , 𝑣
1D NPS [mm2]
• 𝑊 𝑢 𝑊 𝑢, 𝑣 𝑊 𝑢, 𝑣 ∑ 𝑊 𝑢
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DQE
• DQE 𝑢∑
‒ MTF 𝑢 𝑇 𝑢 sinc 𝜋𝑎 𝑢
‒ 𝐹 𝑢 1 T 𝑢, 𝑣 sinc 𝜋𝑎 𝑢
• Assumptions
‒ 𝑚𝛽 100 ⇒ 𝑚𝛽 ≪ MTF 𝑢
‒ Poisson conversion gain from x rays to light quanta ⇒ 𝜀 /𝑚 ≪ 1
• DQE 𝑢∑
‒ No noise aliasing if MTF 𝑢 ≪ 1 for 𝑢 𝑢 1/2𝑥
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Low‐resolution scintillator
• Correlated quantum noise on detector array
‒ System MTF is limited by psf 𝑥 or T 𝑢 not by the detector element size 𝑎‒ T 𝑢 causes the quantum noise in the optical image incident on the photodiode array to be correlated,
which reduces the noise bandwidth
• DQE 𝑢∑
• If T 𝑢 ≪ 1; DQE 𝑢 𝛼
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High‐resolution photoconductor
• Uncorrelated quantum noise on detector array
‒ System MTF is limited by the detector‐array aperture function not by the photoconductor
‒ T 𝑢 is approximately constant over frequencies passed by sinc 𝜋𝑎 𝑢
• DQE 𝑢∑
‒ Note that ∑ sinc 𝜋𝑎 𝑢
• 𝛾 = detector fill factor in the 𝑥 direction
• DQE 𝑢 𝛼𝛾 𝛾 sinc 𝜋𝑎 𝑢
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𝛼 1 𝛾 1.0
0.75
0.5
0.25
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Video‐based portal imaging system
35Bissonnette et al. MP (1997)
a‐Si FPD
36Siewerdsen et al. MP (1997)
Non‐linear response near detector saturation
Image blur
Lag
Zero‐freq. counting statistics
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a‐Se FPD
37Zhao et al. MP (1997); Zhao & Rowlands, MP (1997)